Defect-induced Ferromagnetism in Insulators without Magnetic Ions: The Case of Cation Vacancy in CaO

We have investigated by means of first-principle supercell calculations the posibility of ferromagnetism being induced by cation vacancies in non-magnetic oxides in four steps: ({\em i}) A single neutral Ca vacancy $\mathrm{V_{Ca}^0}$ is found to have a magnetic moment of 1.9 $\mu_{B}$ due to its electronic configuration: $(a_1^2\, t_{1+}^3\, e_+^2)t_{1-}^p\, e_-^q$ where, the $e_-$ state is partially occupied ($q \approx 0.5)$, leading to a transfer of some hole density to the $t_{1-}$ valence band states ($p \approx 2.5)$. ({\em ii}) The ferromagnetic interaction between two vacancies is found to extend only to four neighbors or less. ({\em iii}) To achieve magnetic percolation on a fcc lattice with such an interaction range one needs a minimum vacancy concentration of $1.8 \times 10^{21}$ cm$^{-3}$ (4.9 \%). However, ({\em iv}) due to the high vacancy formation energy even under the most favorable growth conditions one can not obtain at equilibrium more than $10^{18}$ cm$^{-3}$ vacancies. Thus, a non-equilibrium vacancy-enhancement factor of 10$^{3}$ is needed to achieve ferromagnetism in such systems. Comparison with other non-magnetic oxides will be also discussed.